Symmetric Definite Generalized Eigenproblem

1
Symmetric Definite Generalized Eigenproblem

• Problem
• A sym. indefinite and B s.p.d.
• Standard method in Matlab Cholesky-QR
  (optionally QZ)
• Properties
• All eigenvalues should be real
• Eigenvector matrix diagonalizes both A and B
• Easy to solve if A, B well-conditioned

2
Motivation Animation

• Goal create interactive animations of deforming
  objects
• Use finite element method
• Problem solving the PDEs is slow.
• Solution Linear Algebra!
• Kris Hauser, Chen Shen, James OBrien
• Interactive Deformation using Modal Analysis with
  Constraints
• Graphics Interface 2003
• Diagonalize the system to create a set of
  uncoupled differential equations (modes).
• Extract the most important modes from the system,
  and simulate only those.
• The eigenvectors with the largest eigenvalues
  describe the most important modes.
• Perhaps only a dozen, out of thousands, really
  matter.

3
The Eigenproblem

• The FEM model gives us a system of ODEs
• Which can be simplified to use two matrices
• These matrices are symmetric positive definite
• Modal analysis is a generalized eigenproblem

4
Conditioning

• We want to apply this method to any mesh we
  happen to throw at it.
• Some models lead to ill-conditioned matrices.
• Depends on shape of elements

• Figure by Jonathan Shewchuk, from What is a good
  finite element

5
Davies, Higham and Tisseur

• Cholesky on B
• Jacobis method to solve eigenproblem
• Iterative refinement based on Newtons
• Claim new error bound, potentially better

6
Error bounds and numerical results

• Old bound
• Want to get rid of
• New bound
• Where

7

• Newtons method is applied to the equivalent
  problem
• They first present an algorithm for iterative
  refinement, cost O(n3)
• Improved algorithm is O(n2), but at the cost of
  less frequent and less rapid convergence

8
Numerical Results

• n20, AI, BRTR (R is a Kahan matrix)
• Plot of eigenvalues vs. backward error of the
  eigenpairs

9
Chandrasekaran

• Claim new algorithm is numerically stable and
  efficient which satisfies both properties
• All eigenvalues should be real
• Eigenvector matrix diagonalizes both A and B
• Idea Find C such that
• Where and are diagonal
• Eigenvalues
• Eigenvectors

10
Error Bounds and numerical results

• Error bound for an eigenvalue/vector pair
• Numerical experiment
• A and B are 5x5
• Matlabs QZ algorithm gives large imaginary parts
• The algorithm in this paper returns all
  eigenvalues and eigenvectors to full backward
  accuracy